Math Overflow users resolve PhD thesis crisis
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I was in my 6th year. All my friends had graduated, and my stipend had run out. I was 2 weeks away from submission and discovered that one of my assumptions was wrong, which potentially distorted/invalidated all my studies -- to fix these studies would have potentially delayed submission for months. It was a very subtle assumption violation (and it wasn't even that wrong) and my committee probably wouldn't even have noticed. I was tempted to sweep it under the carpet and not let it keep me from graduating.
But I knew it was wrong. I felt that if I sacrificed my integrity then, the moral failure would mark me for life. No one would know -- but I would know. So I decided to fix the issue, re-do the studies and live with the reality that I would have to delay my defense.
Turns out when you're desperate -- and many grad students can attest to this -- a resourcefulness that you never thought you had kicks in ("where were you during all my years of grad school?"). I don't remember how, but I somehow managed to wrangle new studies out in 3 days (which would have previously taken me months). I made the deadline in the end.
The lesson I learned was that committing to doing the right thing has its costs, but in some cases it also forces one to explore attacks never previously considered. Asking on MathOverflow is one such attack.
When people go to grad school, then during their studentship get married and have a kid -- suddenly their productivity goes up dramatically and output per time increases by many factors. What happened?
It turns out when you have self-motivating reasons to get out (and a target on your head from your spouse to earn some money for godssake), you find ways to focus on what's important and drop the rest.
No more idling away for hours on silly ideas that don't get you closer to handing in your thesis. No more trying random libraries that get your code to run 2% faster. No more goofing around after 6pm with other students just because you have the time -- you have to get home and be a breadwinner for your family. You have to get shit done.
You start to ask, "even if I don't know exactly what the thesis will say, how should it be organized and what kinds of conclusions will make up the writing? And what experiments do I need to fill in those charts/paragraphs, and no more?" What's the minimum I need to do to get out of here? Not, "What amazing interesting thing could I explore?"
Limits and constraints sometimes free the mind dramatically. The side effect is maybe you don't get to explore ideas that go nowhere, but that's a discussion about the purpose of the PhD and for another topic I guess.
(And sometimes, if you think, well I don't have a kid, so what's the rush? Well, someday you might have a spouse, a kid, and every day of time you left in grad school is a day for your future self -- and family -- and $$ -- left behind in time. Work to free your future self... now, while you have the time.)
Amazing breakthroughs have high risk, and make it highly likely you'll have a crisis when it doesn't happen.
And what I mean is that, even if you don't know what the answer is going to be at the end of your research, you must think about, or have an idea about, the format of what that amazing answer is going to be. Write the outline of your thesis and "ghost out" what the major charts will be. Write the intro sentences of each chapter -- what are they? (and I don't just mean the boring review of the field part, but your findings part)
You should know what major finding, plot, or table your research is going to output. What are the columns and rows of that table, or axes of that plot? How many data points are required? How many of them can already be guessed? Where is the surprise going to be? What is the conclusion going to be?
For most graduate research, the finding is not an amazing field-changing big bang. Few grad students are that fortunate. Or at least most of the bulk of the work will be of that sort. You should be able to predict what the answer is going to look like from past work, and the error bars.
Draw out the answer you're aiming for, now. If you can't even articuate what the answer will look like, you may be in for a bad time, so work on fixing that. It will also push you and your advisor to be specific about what the output of your thesis will be.
Your future self will thank you.
If you basically know the answer in advance, chances are you are not doing very interesting research.
The worst work I have done has been of this sort. The best has had me completely change my view of (mathematical) reality multiple times during the process.
If you (the general "you") are in industry breakthrough territory, you're an advanced student and my advice isn't for you. Otherwise, I think it's a good practice.
Fair. I think considering what possible conclusions could come is certainly important. I worried about your original post suggesting actually writing the conclusion in advance.
An effective PhD advisor/thesis isn't wandering the woods to find something. It's a guided coaching exercise, with an outcome in mind. If your advisor doesn't know this, maybe time to find a new advisor...
You don't know me, but I really needed to hear this. I left grad school essentially because of a dearth of coaching. Thank you for framing this so succinctly.
There are multiple slightly different versions of the seminar (another is linked below that one), but unfortunately they all came out after I'd finished my PhD, and I didn't hear that advice from anyone else.
The process I arrived at after losing time on failed projects was basically "fail fast": find the simplest quickest way to demonstrate that your idea won't work, and do that. Then find the next simplest, and so on, until either it works or you've proven it doesn't and moved on.
Really annoys me when the word fraud is thrown around so freely, so I'd appreciate it if you don't.
> Obviously you rewrite the whole paper
It is not mentioned and was not obvious to me. Still, I'd postpone writing the conclusion to after you actually have the data. To do otherwise would be steering the experiments to get the expected results. I also reckon this depends on your field, how long the experiments take and your confidence in the results.
> Really annoys me when the word fraud is thrown around so freely, so I'd appreciate it if you don't.
I agree, 'academic dishonesty' is a more appropriate term here.
I remember after my first paper was finished and submitted, and wow, it seemed easy to bang out papers after that, all you needed was four to five figures :)
Does it ever works that way for field changing big bang ideas? Every human achievement was built upon the back of previous achievement and works, either by you or someone else.
Most nobel/turing/fields level work doesn't happen while the researcher is still in graduate school, because a) it usually benefits from additional experience, knowledge, and resources and b) it can sometimes take years to complete.
It also might not be obvious to a dissertation committee (or anyone else) how field-changing certain work actually is until many years have passed, and they might dismiss it early on. Consider Tim Berners-Lee, who was actually a Physicist but received the Turing award because the web (and HTTP/HTML) turned out to have much greater impact than predecessors like FTP, NNTP, PLATO, HyperCard, Director, Xanadu, Intermedia, Gopher, AOL, etc..
Another reason that more mundane research tends to work better is that groundbreaking work is often hard to publish (and get support/funding for), particularly if it isn't completely solid and seems to contradict or supersede established theory or practice.
However, if you are choosing between multiple projects that all seem doable within the allotted time frame, picking the one with higher positive impact on the field (and/or elsewhere) may be a good approach.
Not that I minded, just saying that time constraints and sleep deprivation seemed to have the effect you'd except from them.
High levels of sleep deprivation is only common for the first few months.
> not having the time on nights and weekends to come up to speed on any new technology.
This tends to be a problem on teams that don’t properly evaluate costs of new technology and so churn like crazy for very minor productivity increases.
"I won't do that" can be worth $100k.
I totally get it, but it makes me sad. That line, right there, probably explains why it's so few people who actually get to make the big discoveries.
I credit my now wife with the (passive) motivation to shave at least 6-12 months off my time in grad school.
He understands about 40% as much about the biology, yet he's very good at getting people to drop/trim things that are not going to be worthwhile. He often adds tons of constraints that allow our researchers to hone in on what's going to drive the company forward instead of languishing in a million possibilities.
I very quickly found myself reprioritizing how quickly I could graduate and find a job (since my wife's field didn't have many lucrative career prospects).
That is why I did not do any dating during my PhD
This is getting millions of students through high school, college, and university every year.
Everybody has at least one thing they pushed back for way too long and then developed almost super-human-like abilities to do it. Doesn't guarantee a good grade, though.
It reminds me of a story I heard from a businessman that was walking with an associate down the street. He stopped at one of those old metal newspaper dispenser machines to buy a paper. For those that don't know, the way they work is once you have put in your quarter the machine opens up and all the papers are just stacked in there.
His associate asked him to grab a newspaper for him. So he pulled out one paper, closed the machine and put in another quarter and then pulled out another. The associate asked, "Why didn't you just grab a second paper for free? It's just a quarter." The businessman's response was, "If I'm willing to sacrifice my integrity for a quarter, how much easier will it be to sacrifice it when something serious is on the line?"
> 10 “He who is faithful in a very little is faithful also in much; and he who is dishonest in a very little is dishonest also in much.
> 11 If then you have not been faithful in the unrighteous mammon, who will entrust to you the true riches?
It's right there in the gospels, plain as day, and yet I agree with you that there's far too little thinking like this in the world, even among people who claim to be "Christians".
Why are we not resourceful in normal time?
Calvin: No, I'm waiting for inspiration. You can't turn on creativity like a faucet. You have to be in the right mood.
Hobbes: What mood is that?
Calvin: Last-minute panic.
Yeah he burned out quickly and left with a masters.
There are real quality differences between the two scenarios, but they aren't nearly as bad as what you'd think they are. The reality is that I can just work a lot more efficiently when stressed. If I could do that all the time I'd be able to work 3 hours a day and remain as productive.
Funny because he was often the reason we had to do the last minute rush :)
I wrote a special chapter where I discussed my results with myself and ended up suggesting someone takes a closer look if interested.
The jury was initially 6 people, 1 refused to review a thesis where the candidate discusses with himself because it breaks tradition. The others were quite happy, with one of them referring to my discussion and thanking me for the time she gained with it and which allowed her to spend more time with friends.
For some small issues it is sometimes better to tell it upfront and highlight hiw genius the rest of the thesis is.
> From OP's point of view this could be viewed as glass half-full rather than glass half-empty. Their dissertation results hold unequivocally on the sphere and might hold on the torus, though it is an open problem if they do. It is certainly legitimate to study what follows from a given conjecture being true. It could even be spun as a feature rather than a bug of the dissertation. If the results in fact fail on the torus then you know that the conjecture must be false. Potentially, it could open up a fruitful avenue of attack.
Kind of reminds me of Terence Tao's post on what solving big problems looks like: https://terrytao.wordpress.com/career-advice/be-sceptical-of...
It was only later that I found out that history, the study of art & literature, and philosophy can all teach you the same thing. The important part is that you're interested in the topic.
As a student representative for my undergraduate mathematics course, I got really pissed off at lecturers for exactly this reason: they'd write out a perfect correct proof on the whiteboard, but wouldn't explain where it had come from or how people had arrived at the solution. We were left to figure that out on our own.
They then complained that students were rote-learning for exams, rather than coming to a full understanding of the material. I'm not sure what they were expecting, given that that's exactly how they were teaching it.
He has an identical twin brother, who is also a math professor at the same college. The regular professor was out for a day, and his brother came in to teach the class. His teaching style was completely different. "Ok, we need to prove X. Where should we start?" and would sit on the table and look at us with an inquisitive look on his face. Then learning happened.
Everyone's mind was blown. Most people didn't realize it was a different person. Then on Thursday it was back to same-old same-old.
I still don't know shit about discrete math.
I ask them such a question, and then wait for 15 minutes, and no one says anything.
Proofs in math journals are given 'as-is' and you learn the intuition through social means and discussions.
Anyone could modify the proofs and add comments/explanations.
This is an excellent metaphor using beautiful language. Did you just coin this yourself or is it pre-existing?
I suspect the reason OP's thesis worked out okay is because his intuition wrt the problem is correct, even if his formulation was a bit off. Very cool, sounds like a good mathematician to me
Stack overflow and it's cousin sites have many serendipities like this - and I happily conjecture this happens more here than facebook or twitter.
I think the reason is that despite being a walled garden (ie proprietary) it still has a promise to open up the content and makes effort to moderate and grow the community - in other words what they are really selling is not the SEO but the sweet spot between "anyone posts anything" of an "ideal" internet where no rentiers exist but no one can find anyone else, and the much more corporate hand of Facebook.
I am not sure reddit exists in this sweet spot either - mostly because there is just sooo much reddit.
- The site is operated by Stack Overflow/Exchange, but is owned by MathOverflow, Inc a non-profit corporation[0]. As such, it retains the right to exist independently of the Stack Overflow company - to my knowledge, it is the only public Stack Exchange site for which this is true.
- Like all public Stack Exchange sites, authors retain ownership of their work, which is published under a CC-BY-SA license. Regular archives are uploaded to Archive.org and can be obtained there or via Bittorrent[1]
In short, not a walled garden, and not Stack Overflow's garden.
[0]: https://meta.mathoverflow.net/questions/969/who-owns-mathove...
Yes, I think I am wrong to use the term walled garden, but it's hard to think of something else.
In a "platonic ideal" of the internet everyone would have their own internet connection, and a server and say post their own interesting queries and somehow others would find and answer them.
Perhaps search was assumed to solve it all then.
But the universe is much more "clumpy" than that so people will gather around certain locations, in nature they are natural oasis.
Perhaps we should drop the walled garden idea - gardens, walled or otherwise need tending and upkeep and that passed the ability of one or two people to do in their spare time somewhere around 1991 on usenet.
Tending a garden is a costly affair.
I think perhaps walled city might be a better term? It implies the "never leaving" which is what facebook seems to aspire to.
perhaps a better analogy is "chargeable car parking". :-)
The antonym to the walled garden is the open garden or field, with hardy plants able to withstand and even thrive in the local atmosphere. They're still cultivated - weeded, fertilized - but there's no need to create a microclimate to just to accommodate them.
In this context, Facebook does make some effort toward walling off their gardens, but... As you note, Facebook's primary goal isn't protecting it's dominating - Facebook is just as happy to own major portions of the 'Net in pursuit of this goal, and more than a little reluctant to provide any real protection beyond what is absolutely necessary.
Beyond that... We all garden. From little personal websites and blogs, to big community gardens[1] like Wikipedia, Reddit, Stack Overflow, and even Hacker News. We plant, we harvest, we tend these plots, alone or together, but make little effort to isolate them from the larger world - indeed, we generally recognize that the strength of the Web is based on its interconnections, its inherent ability to draw together different sources of information.
[0]: https://www.orwell.ru/library/essays/politics/english/e_poli...
[1]: https://meta.stackexchange.com/questions/349513/feedback-for...
Neither definition actually makes much sense when talking about incompatible protocols.
Guess the one above this poster has right that the analogy is not totally right. Still if one assume the flower can be viewed a trillion time but the question is creation, walled garden is a good metaphor.
I still see the walled garden analogy to be more useful
If people are confusing their definition of a metaphor and mine (I don't think Orwell mentioned that but maybe it's low down the list of stylistic errors) then that can cause problems. But I struggle to see how Facebook / twitter "protects" creators from the harsh winds of the outside.
But... That sense is dying; it is a poor metaphor because few people actually build real-world walled gardens[0][1] for that purpose; the metaphor has no currency, and thus the meaning shifts. Now it is just as frequently used to describe any sort of system which restricts the flow of people OR of information, in or out. So while Instagram might be considered a "walled garden" in the original sense (no outbound links on posts), it may ALSO be considered a "walled garden" in the sense that it restricts inbound access for non-members, or even in the sense that it forces certain onerous licensing terms on contributors. In this manner, the metaphor becomes problematic, as what one intends to convey is not necessarily the meaning which is understood by others. If/when the metaphor dies entirely, becoming an idiomatic way of saying "not completely open", this problem disappears - no one will attempt to relate people to plants, or content to flowers.
With this in mind, I suspect your original intent was focused on the "garden" aspect: that the value MO provides comes from imposing a structure and certain expectations which facilitate productive interactions like this, with the "sweet spot" being that it remains open enough for the rest of us to benefit from the outcome of those interactions.
[0]: https://solar.lowtechmagazine.com/2015/12/fruit-walls-urban-...
That is how it used to be
I was late to the party, but I had my own website with a guestbook around 20 years ago (well hosted by AOL, but you would not know that through the url redirection)
As a private company, probably someday they will lost their techno-utopian magic (as Google already lost). It will be a very sad day.
If your reader already knows most of the context and has read the previous posts, then "re-posts" are bad, long-running threads are good, and they just want to see the latest updates.
If your reader is coming to the material cold, a "re-post" may be completely new to them, and posts that presume pages and pages of existing context are completely impenetrable. You want to lean towards fresh, short threads.
The challenge for designing a forum then is balancing the competing needs of those users. You can make some progress by tracking on a per-user basis which comments they've already seen. Reddit does that (maybe just for gold accounts?) where new comments are blue. That makes it pretty easy to skim through a comment thread and see just the new leaves.
But there's still the question of how to sort the threads themselves. It might be interesting if that sorting was also user-specific. Maybe deprioritize threads that you've seen but not interacted with, and prioritize "live" threads that you've participated in and are still changing.
"Twitter is a few people saying interesting things amidst a much larger number saying mean or mistaken things. So are books. But you don't suddenly get sentences from bad books in the middle of reading one of the good ones. Maybe this gets fixed in version 2 of social media. Maybe version 2 is halfway between the randomness of Twitter and the predictability of Substack."
I’m not sure what sort of advertising they do on other stack exchange sites if any, but sticking to a job board on SO makes it so much more pleasant to read than a social feed throwing random junk products at you every few minutes until you go away.
But I do recall a similar confusion happening with "piecewise linear" (the question is whether the pieces have to fit together).
Anyone is research math does consider the domain and target to be part of the function. Alas, that doesn't mean that they'll actually write down which domain and target they have in mind.
"A function or curve is piecewise continuous if it is continuous on all but a finite number of points at which certain matching conditions are sometimes required."
Emphasis added.
The problem is that the students of the first (or second) year of the university will then try to use the the Bolzano's theorem / Intermediate value theorem to prove that it has a zero in the interval [-1, 1].
So I must answer NO, but the problem is that usually in the question the domain is implicit, so it cause a lot of confusion.
I think it's more clear the definition of "piecewise differentiable". I'm not sure what is the "official" definition of "piecewise continuous". The definition in Mathworld looks a little fuzzy. I'd probably request not an horrible behavior in the borders of the intervals that are glued, like in
* sign(x) -> yes
* 1/x -> no
* sin(1/x) -> no
One of the best keep secrets in math is that the definitions are somewhat arbitrary.
[1] At least it is how we call it in Argentina, sometimes the names/definitions change in each country.
f is maybe (piece-wise) continuous over what pseudo-domain? R or R\{0}?
You could axiomitize that an infinite discontinuity is like an unbouned function as x->infinity, but then how would you avoid 1/x being regular continuous?
I think you are claimokg that a set being incomplete "at the end" is different from a set having a hole in the middle -- that the question of continuity presumes connected sets. That's not standard but might be an appropriate assumption for the context of your paper.
Anyway, arguing over terminology is boring unless it raises conceptual issues. The point is to communicate, which has at least 2 stakeholders. Clarify your terms and move on.
The domain is R in this case. The less-restrictive definition would be that f is piece-wise continuous if there are a discrete set of points .. < x_0 < x_1 < ... with f continuous on each interval (x_i, x_{i+1}). The alternative definition is that, plus requiring that the restriction of f to those intervals have limits at the endpoint. For piecewise smooth function, there's an even larger variety of possible meanings, yet it's often stated without clarification.
This sounds kind of surreal; putting them in shorter terms, we have two rival definitions for "piecewise continuous":
1. A function f is piecewise continuous if there exist one or more intervals over which f is continuous.
2. A function f is piecewise continuous if there exist one or more intervals over which f is (1) continuous, and (2) bounded.
I agree that it sounds obvious which of those is more appropriate as a definition of "piecewise continuous"...
(It also worries me that the definition you give requires the intervals to be adjacent, but doesn't require that more than one interval exist. A function that is continuous over (-2, -1) and also over (1, 2), but not anywhere else, would meet this definition, but you wouldn't be able to include both of those intervals of continuity in the set of endpoints.
I would prefer to either have two sets of endpoints, such that we end up saying f is continuous over (x_i, y_i), (x_{i+1}, y_{i+1}), etc. (if we want to allow for intervals of discontinuity), or to say that the intervals (-inf, x_0) and (x_n, +inf) also count (if we don't).
However, if we take that second approach, and we go with the definition of piecewise continuity that requires the function be bounded over every interval, we've just defined functions like f(x) = x as being not piecewise continuous despite the fact that they are continuous.)
And saying left and right limits is -infinity or +infinity really also isn't that weird. I'm pretty sure other metric spaces can be likewise extended and end up with similar algebraic laws as the "extended real numbers". Again this isn't very profound, but is good for education and efficient communication, and so should be perused.
Finally, it's interesting that measure spaces with infinite measure is already a thing that people. I would like to see more connections with metric and measure spaces; e.g. we can have an n-point metric which is defined using the measure of the (n-1)-simplex. Just as regular metric spaces have a "triangle inequality", 3-point metric spaces should have a "tetrahedron inequality", and n-point metric spaces should have a "(n-1)-simplex inequality". Again, this is not profound, but good for communication, and connections between definitions help one compress everything for better mental storage.
As a STEM grad student (not in math), I had more than a couple such moments of crises, when I posted my questions on various stackexchange websites. I got either useless replies, or no replies.
No responsible advisor would let the work get to such a state, so late in the game. Major fault of the advisor too, here.
The last year of my PhD I ended up being pretty much alone because my advisor had changed research topics a year before and therefore was not interested nor up to date, so any of her inputs were not very useful.
A couple friends of mine also struggled with their advisor because he actively avoided communication for some reason. I guess he had a personal or health issue.
So even on good faith, advisors can end up making students life quite stressful for one reason or another
As far as I could tell, the analysis was already done -- but my samples were needed for some other reason. I was kind of bemused by the idea that the analysis would be invalid with nothing behind it, but valid with unrelated data behind it.
There was no interesting novel outcome: he was random-sampling prior states of memory.
I felt very bad for him, it was mid-stage. I didn't hear how he resolved it.
The other side of this is the crisis which only emerges in the viva. I was working in Leeds uni in the 80s and overheard an external discussing a case he had: it was obvious the results were fraud. They made the student and his supervisor to the sums in the room, on the blackboard. He didn't get his thesis.
Although now I have finished the thesis without that part (it should have become an additional chapter). Perhaps I should post it around (although that might spoil it for a paper)
Consider n polynomial equations in variables x_1, .., x_n, with constants a_1,..,a_n, b_1,..,b_n, c_1,...,c_n:
p_1 := a_1 x_1 x_2 + b_1 x_1 + c_1 x_2 + d_1 = 0
...
p_{n-1} := a_{n-1} x_{n-1} x_n + b_{n-1} x_{n-1} + c_{n-1} x_n + d_n = 0
p_n := a_{n} x_{n} x_1 + b_{n} x_{n} + c_{n} x_1 + d_n = 0
I have found a recursive approach that results in a quadratic equation, containing only a single variable x_i (and the constants). (It is too much for a comment, here is a PDF: http://benibela.de/tmp/quadratic-equations-recursion.pdf )
For example for n = 2, it is very simple: x^2_1 (a_2 b_1 - a_1 c_2) + x_1 (a_2 d_1 + b_2 b_1 - a_1 d_2 - c_1 c_2) + b_2 d_1 - c_1 d_2
This gives 2 solution. But I do not know what happens if the terms cancel each other out. Like if a_2 b_1 - a_1 c_2 = 0, there would only be one solution. But since the full solution in the pdf is so complex, I do not see which constraints would lead to cancellation there.
---
And that is not the full problem I was trying to solve. In the full problem there are constraints on the a, b, c, d. There is a given graph, and depending which nodes are connected in the graph, the constants are the same. Like if node 3 and node 7 are connected, then b_3 = c_7 und c_7 = b_3. (even more complex though). And then the question is, do these constants cancel in the solution of those equations? And the final problem we want to solve: which graphs lead to exactly one solution, and which graphs lead to no solution of the equations?
the hell does that mean?
Ambiguous and poorly explained. (Note the question immediately afterwards asking for clarification.) But probably something along the general lines of "My advisor said that, if my main theorem is an asymptotic estimate instead of an exact formula, then this would not be judged to be novel/strong enough to earn a Ph.D."
But roughly speaking, imagine you have two functions f(t) and g(t), which are described in completely different ways, and you want to prove that f(t) = g(t). If you try and fail, then you might instead aim for a proof that the difference between f and g is bounded, or that f(t) = O(g(t)) and vice versa, or that the limit of the ratio between f and g is 1, or something along these lines.
In many cases, such partial results are also of interest. In general, partial successes in math are considered to be successes.
But in some cases, partial results aren't really considered all that interesting -- or perhaps are known already or can be obtained very easily.
Note, another implementation doesn't have this behaviour for the family of inputs I use. It's an implementation detail that has no effect on correctness. Thus for the other implementation another family should exist.
The sentence did strike me a little wierd though.
I knew I could take this route but never did it. This is a bit of a mockery of the whole purpose of a PhD thesis (it has to be your work primarily).
This is just an advanced version of posting homework questions on the internet.
But if you dig a bit deeper, I can see two counter-arguments:
1. The real risk -- by which I mean "the risk I have most often observed in the wild" -- is that a Ph.D expands to fill the time it's given, without ever wrapping up and producing a publishable result. This happens so often that it's practically expected in some places.
2. Having a deadline, oddly enough, also serves as a catalyst for birthing an idea... for "pinching it off" as the expression goes. At some point you have to stop planning and start executing. You can see the deadline as a forcing function.
Ph.Ds are needlessly traumatic and procedural in many ways, but I'm no longer sure that hard deadlines are a net negative.
I feel like this wisdom isn't tapped into enough. We're often burdened with individual tasks and challenges while utilizing crowd knowledge is looked down upon or seen as an inferior solution finding mechanism. e.g. Imagine if companies worked together to figure out self-driving cars rather than compete?